The present invention relates to a mass spectrometer and a method of mass spectrometry.
U.S. Pat. No. 5,783,824 discloses a linear ion trap wherein an axial quadratic electrostatic potential is superimposed along the length of the ion trap. Ions are ejected axially from the ion trap by resonance excitation wherein a supplementary excitation axial potential is applied to electrodes of the ion trap. The supplementary axial potential has a frequency which corresponds with the fundamental harmonic frequency of the ions which are desired to be ejected.
The equation of motion for a forced linear harmonic oscillator is given by:
                                          z            ¨                    +                                    q              m                        ·            k            ·            z                          =                              -                          q              m                                ⁢          a          ⁢                                          ⁢                      cos            ⁡                          (                              σ                ·                t                            )                                                          (        1        )            wherein k is the field constant of the axial quadratic potential (see Eqn. 27), a is the field constant of the modulated axial potential, σ is the frequency of modulation of the axial potential and q is the electron charge multiplied by the number of charges on the ion and m is the molecular mass of the ion.
The exact solution is given below:
                              z          ⁡                      (            t            )                          =                                            z              1                        ⁢                          cos              ⁡                              (                                  ω                  ·                  t                                )                                              +                                                    (                                  2                  ·                                      V                    /                    k                                                  )                                      ·                          sin              ⁡                              (                                  ω                  ·                  t                                )                                              +                                                    q                ·                k                ·                a                                            m                ⁡                                  (                                                            ω                      2                                        -                                          σ                      2                                                        )                                                      ⁡                          [                                                cos                  ⁡                                      (                                          σ                      ⁣                                              ·                        t                                                              )                                                  -                                  cos                  ⁡                                      (                                          ω                      ·                      t                                        )                                                              ]                                                          (        2        )            wherein z1 is the initial z coordinate of the ion at t=0 and V is the initial potential of the ion in the z direction at t=0.
Furthermore:ω=√{square root over (q·k/m)}  (3)wherein ω is the frequency of simple harmonic motion of the ion in the axial electrostatic field.
The amplitude of oscillations depends upon the driving frequency σ. The amplitude of oscillations has its maximum when the driving frequency matches the fundamental harmonic frequency ω. Under these conditions the system undergoes resonant excitation.
Eqn. 1 describes the situation where the excitation waveform has a linear potential gradient. The field is uniform in space and changes direction or amplitude with time. More generally excitation will be dominated by resonance at the fundamental harmonic frequency when the excitation waveform is of a form which may be expressed by the general series expansion:
                              V          ⁡                      (            t            )                          =                              cos            ⁡                          (                              σ                ⁢                                                                  ⁢                t                            )                                ·                                    ∑                              n                =                0                            ∞                        ⁢                                          C                n                            ⁢                              z                                  (                                                            2                      ⁢                      n                                        +                    1                                    )                                                                                        (        4        )            where n is an integer number n=0 . . . ∞, Cn is a coefficient for each order term and σ is the frequency of modulation of the supplementary axial excitation potential.
For example, for dipolar resonance excitation in a Paul ion trap and RF quadrupole devices, it can be seen that the periodic term in Eqn. 1 corresponds to n=0 in Eqn. 4 with C0=a.
According to the arrangement described in U.S. Pat. No. 5,783,824 axial ejection of ions occurs when the frequency of modulation is substantially equal to the fundamental harmonic frequency of ion oscillation. However, this approach has been found to suffer from a relatively low mass resolution given a fixed rate of scanning of the excitation frequency or the depth of the electrostatic potential well.